Find the velocity, speed, and acceleration, and describe the motion of a particle whose position at time t is:
$r = 3\cos(\omega t)i+4\cos(\omega t)j + 5\sin(\omega t)k$
They then note in regards to the motion:
Observe $|r| = 5$. Therefore, the path of the particle lies on the sphere with equation $x^2 + y^2 + z^2 = 25$. Since $x=3\cos(\omega t)$ and $y = 4\cos(\omega t)$, the path also lies on the vertical plane $4x = 3y$.
I don't understand the reasoning behind the sphere. I rely much on the principles used in order to trace a graph for 3-dimensional, i.e the general equations for different parabloids and cylinders in 3d.
So I'm not sure how to trace with coordinates $ x=3\cos(\omega t)$,$ y = 4\cos(\omega t)$ and $ z = 5\sin(\omega t)$.
How does the magnitude of $r$ determine that the particle lies on sphere? I cannot reconcile this with previous literature.